Hi Manos
If I understand you correctly your wanting to count for each person how many dates are before today (eg how many tasks haven't completed by the due date).

If so then this formula will do that:
=COUNTIFS(Φύλλο1!$A$2:$A$16,A2,Φύλλο1!$B$2:$B$16,"<"&$D$1)Enter this in Sheet2 cell A2, and drag down. As it stands, if the date = TODAY() then it won't be counted.